Problem 42 - Precalculus Essentials

Question

Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$f(x)=-3 x+2$$

Step-by-Step Solution

Answer

The slope of the line given by the function \(f(x) = -3x + 2\) is -3 and the \(y\)-intercept is 2. The graph of this function is a straight line, decreasing from left to right, passing through the point (0, 2) on the \(y\)-axis.

Step 1: Identify the slope

The slope 'm' can be directly identified from the equation. Here in the equation \(f(x) = -3x + 2\), the coefficient of 'x' is the slope. Therefore, the slope of the line is -3.

Step 2: Identify the \(y\)-intercept

The \(y\)-intercept 'c' is the constant term in the equation. In the equation \(f(x) = -3x + 2\), the constant term is 2. Therefore, the \(y\)-intercept of the line is 2.

Step 3: Graph the function

To graph the function \(f(x) = -3x + 2\), start with plotting the \(y\)-intercept at '2' on the \(y\)-axis. The slope of the line is -3, which means for every unit increase in \(x\), \(y\) decreases by 3 units. From the \(y\)-intercept, move to the next point by going one step right and three steps down. Connect these points with a straight line to complete the graph. The direction of the line should be decreasing from left to right since the slope of the line is negative.

Key Concepts

Slope

The slope is a key concept when it comes to understanding linear equations and functions. It tells us how steep a line is on a graph and also indicates the direction in which the line moves. If you're working with a linear equation in the form of \(y = mx + c\), the slope \(m\) is the number in front of \(x\). This number is crucial because it affects how the line tells its story on a graph.

The slope can be:

• Positive: This means the line rises as you move left to right.
• Negative: This means the line falls as you move left to right.
• Zero: This results in a horizontal line.
• Undefined: An undefined slope is associated with vertical lines.

In the given exercise, the slope is \(-3\). This tells us that for every step you take to the right on the x-axis, you need to move three steps down on the y-axis. A negative slope confirms that the line decreases from left to right.

Y-Intercept

The y-intercept is another fundamental concept in graphing linear equations. This is the point where the line crosses the y-axis, and it's often symbolized by the letter \(c\) in the equation \(y = mx + c\). Understanding the y-intercept helps you pinpoint the precise starting point of a line on a graph.

Some key features of the y-intercept:

• It usually has the coordinate \((0, c)\), because when \(x = 0\), \(y = c\).
• It provides the starting point from which you'll plot your line with the help of the slope.

For the equation \(f(x) = -3x + 2\), the y-intercept is \(2\). This means the line will cross the y-axis at the point \((0, 2)\). Plotting this point on the graph gives you a starting location for the line that is very important for the next step, which is graphing the entire function.

Graphing Linear Functions

Graphing linear functions is an approachable process once you understand the roles of both the slope and the y-intercept. It's all about plotting these points and connecting them to reveal the line that represents the equation.

Here's how you can graph a linear function step-by-step:

• Start Plotting the Y-Intercept: Always begin by marking the y-intercept on the graph. It's your anchor point.
• Use the Slope: From the y-intercept, use the slope to find the next point. If the slope is \(-3\), as in \(f(x) = -3x + 2\), you would go 3 steps down and 1 step to the right.
• Draw the Line: After plotting a second point, draw a straight line through the points. Extend the line in both directions to accurately represent the equation.

In our exercise, we take the y-intercept (2) and plot it, then apply the slope (-3) to find another point. Connect these dots, and you'll see why the whole line dips downward—a great visual of how the slope interacts with the graph.